The pith of the present work is to formulate a mathematical model to study the effect of inhomogeneity on dispersion curves of Rayleigh-type wave (RTW) propagating through an imperfectly bonded stratified structure composed of unlike pre-stressed slightly compressible elastic materials. Geometrically, the stratified structure encompasses inhomogeneous pre-stressed slightly compressible elastic stratum (IHPSSCEST) concatenated to another inhomogeneous pre-stressed slightly compressible elastic substrate (IHPSSCESB) with an imperfect interface. Analytical techniques incorporating variable separable approach along with substitution method, are applied to derive closed form expressions of displacement components and dispersion relation of RTW. The derived results are further deduced as special cases, that concur well with the established results in literature. Numerical computation pertaining to the derived dispersion relation of RTW in the considered geometrical model is carried out, and the impacts of inhomogeneity parameters, wave number, hydrostatic stress, and normal and tangential interfacial imperfect parameters on RTW velocity are manifested graphically. The computational study imparts that RTW velocity is prominently influenced by the presence of inhomogeneity in the stratified structure. On applicative grounds, this study may be useful for vibration isolation, shock absorber, engine mounts, flex-joints and so forth in civil, mechanical and offshore structures.